Abstract

We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the homogeneous unsynchronized phase, where the coupled dynamics is in a chaotic unsynchronized state, but exhibits qualitative similar symbolic dynamics at all the nodes in the network. We refer to this dynamical behavior as symbolic synchronization. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behavior of the largest Lyapunov exponent and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local chaotic dynamical function. We present results for two modeldynamics,iterations of the one-dimensional logistic map and the two-dimensional Hénon map, as local dynamical function.

Received 06 April 2006Accepted 20 July 2006Published online 13 September 2006

Lead Paragraph: Nonlinear dynamical elements interacting with each other can lead to synchronization or other types of coherent behavior at the system scale. Coupled map models are one of the most widely accepted models to understand these behaviors in systems from many diverse fields such as physics, biology, ecology, etc. Their important feature is that the individual elements can already exhibit some complex behavior, for example, chaotic dynamics. The question then is how to detect coordination at larger scales beyond the simplest one, synchronization. An important tool in the analysis of dynamical systems are symbolic dynamics. We develop a new scheme of symbolic dynamics that is based on the special partitions of the phase space that prevent the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we report a new behavior of coupled dynamics, which we refer to as symbolic synchronization, i.e., synchronization of the nodes at the coarse grained level, whereas microscopically all elements behave differently. Through the framework of this symbolic dynamics, we detect various global properties of coupled dynamics on networks by using a scalar time series of any randomly selected node. A decisive advantage of our method is that the global properties are inferred by using a short time series, hence the method is computationally fast, does not depend on the size of the network, and is reasonably robust against external noise.